Quadratic refinements (awaiting review)

A quadratic refinement of a modular curve $X_H$ is a modular cover $X_{H'} \to X_H$ such that $H'<H$ and $H=\{\pm 1\}H'$, so $(H:H')=2$. We also refer to $H'$ as a quadratic refinement of $H$. In particular, if $-1 \not \in H$ then there are no quadratic refinements.

A quadratic refinement $X_{H'} \to X_H$ is an isomorphism of the underlying modular curves, but refines the moduli problem from $H$ to $H'$.